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Lecture 4.5: POSets and Hasse Diagrams

Lecture 4.5: POSets and Hasse Diagrams. CS 250, Discrete Structures, Fall 2011 Nitesh Saxena * Adopted from previous lectures by Cinda Heeren. Course Admin. HW4 has been posted Covers the chapter on Relations (lecture 4.*) Due at 11am on Nov 16 (Wednesday)

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Lecture 4.5: POSets and Hasse Diagrams

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  1. Lecture 4.5: POSets and Hasse Diagrams CS 250, Discrete Structures, Fall 2011 NiteshSaxena *Adopted from previous lectures by CindaHeeren

  2. Course Admin • HW4 has been posted • Covers the chapter on Relations (lecture 4.*) • Due at 11am on Nov 16 (Wednesday) • Also has a 10-pointer bonus problem • Please start early Lecture 4.5 -- POSets and Hasse Diagrams

  3. Final Exam • Thursday, December 8,  10:45am- 1:15pm, lecture room • Heads up! • Please mark the date/time/place • Our last lecture will be on December 6 • We plan to do a final exam review then Lecture 4.5 -- POSets and Hasse Diagrams

  4. Outline • Hasse Diagrams • Some Definitions and Examples • Maximal and miminal elements • Greatest and least elements • Upper bound and lower bound • Least upper bound and greatest lower bound Lecture 4.5 -- POSets and Hasse Diagrams

  5. 4 3 2 Draw edge (a,b) if a b Don’t draw up arrows Don’t draw self loops Don’t draw transitive edges 1 Hasse Diagrams Hasse diagrams are a special kind of graphs used to describe posets. Ex. In poset ({1,2,3,4}, ), we can draw the following picture to describe the relation. Lecture 4.5 -- POSets and Hasse Diagrams

  6. 111 110 101 011 100 010 001 000 Hasse Diagrams Have you seen this one before? String comparison poset from last lecture Lecture 4.5 -- POSets and Hasse Diagrams

  7. Maximal and Minimal Reds are maximal. Blues are minimal. Consider this poset: Lecture 4.5 -- POSets and Hasse Diagrams

  8. Maximal and Minimal: Example Q: For the poset ({2, 4, 5, 10, 12, 20, 25}, |), what is/are the minimal and maximal? A: minimal: 2 and 5 maximal: 12, 20, 25 Lecture 4.5 -- POSets and Hasse Diagrams

  9. Did you get it right? Intuition: If a is maxiMAL, then no one beats a. If a is maxiMUM, a beats everything. Must minimum and maximum exist? Only if set is finite. No. Only if set is transitive. Yes. Least Element and Greatest Element Definition: In a poset S, an element z is a minimum (or least) element if bS, zb. Write the defn of maximum (geatest)! Lecture 4.5 -- POSets and Hasse Diagrams

  10. Maximal and Minimal: Example Q: For the poset ({2, 4, 5, 10, 12, 20, 25}, |), does the minimum and maximum exist? A: minimum: [divisor of everything] No maximum: [multiple of everything] No Lecture 4.5 -- POSets and Hasse Diagrams

  11. A Property of minimum and maximum Theorem: In every poset, if the maximum element exists, it is unique. Similarly for minimum. Proof: Suppose there are two maximum elements, a1 and a2, with a1a2. Then a1  a2, and a2a1, by defn of maximum. So a1=a2, a contradiction. Thus, our supposition was incorrect, and the maximum element, if it exists, is unique. Similar proof for minimum. Lecture 4.5 -- POSets and Hasse Diagrams

  12. {a, b} has no UB. Upper and Lower Bounds Defn: Let (S, ) be a partial order. If AS, then an upper bound for A is any element x  S (perhaps in A also) such that  a  A, a  x. A lower bound for A is any x  S such that  a  A, x  a. Ex. The upper bound of {g,j} is a. Why not b? a b Ex. The upper bounds of {g,i} is/are… A. I have no clue. B. c and e C. a D. a, c, and e c d e f j h g i Lecture 4.5 -- POSets and Hasse Diagrams

  13. {g, h, i, j} has no LB. Upper and Lower Bounds Defn: Let (S, ) be a partial order. If AS, then an upper bound for A is any element x  S (perhaps in A also) such that  a  A, a  x. A lower bound for A is any x  S such that  a  A, x  a. Ex. The lower bounds of {a,b} are d, f, i, and j. a b Ex. The lower bounds of {c,d} is/are… A. I have no clue. B. f, i C. j, i, g, h D. e, f, j c d e f j h g i Lecture 4.5 -- POSets and Hasse Diagrams

  14. Least Upper Bound and Greatest Lower Bound Defn: Given poset (S, ) and AS, x  S is a leastupper bound (LUB) for A if x is an upper bound and for upper bound y of A, x  y. x is a greatestlower bound (GLB) for A if x is a lower bound and if y  x for every lower bound y of A. a b Ex. LUB of {i,j} = d. Ex. GLB of {g,j} is… A. I have no clue. B. a C. non-existent D. e, f, j c d e f j h g i Lecture 4.5 -- POSets and Hasse Diagrams

  15. This is because c and d are incomparable. LUB and GLB Ex. In the following poset, c and d are lower bounds for {a,b}, but there is no GLB. Similarly, a and b are upper bounds for {c,d}, but there is no LUB. a b c d Lecture 4.5 -- POSets and Hasse Diagrams

  16. Another Example • What are the GLB and LUB, if they exist, of the subset {3, 9, 12} for the poset (Z+, |)? • What are the GLB and LUB, if they exist, of the subset {1, 2, 4, 5, 10} for the poset (Z+, |) Lecture 4.5 -- POSets and Hasse Diagrams

  17. Another Example • What are the GLB and LUB, if they exist, of the subset {3, 9, 12} for the poset (Z+, |)? LUB: [least common multiple] 36 GLB: [greatest common divisor] 3 • What are the GLB and LUB, if they exist, of the subset {1, 2, 4, 5, 10} for the poset (Z+, |) LUB: [least common multiple] 20 GLB: [greatest common divisor] 1 Lecture 4.5 -- POSets and Hasse Diagrams

  18. Example to sum things up For the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |), find the following: • Maximal element(s) • Minimal element(s) • Greatest element, if it exists • Least element, if it exists • Upper bound(s) of {2, 9} • Least upper bound of {2, 9}, if it exists • Lowe bound(s) of {60, 72} • Greatest lower bound of {60, 72}, if it exists Lecture 4.5 -- POSets and Hasse Diagrams

  19. Example to sum things up For the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |), find the following: • Maximal element(s) [not divisors of anything] 27, 48, 60, 72 • Minimal element(s) [not multiples of anything] 2, 9 • Greatest element, if it exists [multiple of everything] No • Least element, if it exists [divisor of everything] No • Upper bound(s) of {2, 9} [common multiples] 18, 36, 72 • Least upper bound of {2, 9}, if it exists [least common multiple] 18 • Lower bound(s) of {60, 72} [common divisors] 2, 4, 6, 12 • Greatest lower bound of {60, 72}, if it exists [greatest common divisor] 12 Lecture 4.5 -- POSets and Hasse Diagrams

  20. More Theorems Theorem: For every poset, if the LUB for a set exist, it must be unique. Similarly for GLB. Proof: Suppose there are two LUB elements, a1 and a2, with a1a2. Then a1  a2, and a2a1, by defn of LUB. So a1=a2, a contradiction. Thus, our supposition was incorrect, and the LUB, if it exists, is unique. Similar proof for GLB. Lecture 4.5 -- POSets and Hasse Diagrams

  21. Today’s Reading • Rosen 9.6 Lecture 4.5 -- POSets and Hasse Diagrams

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